Problem Set 06
Note:
The following problem set is due October 17 by midnight. Please return directly to
me in my oFce Rutherford 321. If I’m not there slip your assignment below my door.
1.
Let
M
be the matrix representation of an operator in a basis of a Hilbert space spanned
by
k
orthonormal vectors. The matrix
M
is a particular nondiagonal matrix in this space
with
k
eigenvalues
²
k
such that
²
k
<<
1. Show that the following determinant identity:
p
det (1 +
M
)
=
∞
X
m
=0
1
2
m
m
!
"
tr
±
∞
X
n
=1
M
n
n
²
#
m
holds no matter what representation of the operator
M
we want to study. The symbol tr
denotes the trace of sum of
n
powers of the matrix
M
.
2.
Give short answers to the following questions:
(a) Consider a Hilbert space spanned by
j
=
3
2
states. Determine the matrix representation
of any generic operator
F
(
J
x
) in this space given that
F
(
x
) is an analytic function of
x
.
(b) A Hilbert space is spanned by 2
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 Fall '05
 KeshavDasgupta
 mechanics, Hilbert space, Pauli Matrix

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